Differential and Integral Equations

The Dirichlet problem for some nonlocal diffusion equations

Emmanuel Chasseigne

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Abstract

We study the Dirichlet problem for the non-local diffusion equation $u_t=\int\{u(x+z,t)-u(x,t)\}{\,\mathrm{d}\mu}(z)$, where $\mu$ is a $L^1$ function and $``u=\varphi$ on $\partial\Omega\times(0,\infty)$'' has to be understood in a non-classical sense. We prove existence and uniqueness results of solutions in this setting. Moreover, we prove that our solutions coincide with those obtained through the standard ``vanishing viscosity method'', but show that a boundary layer occurs, the solution does not take the boundary data in the classical sense on $\partial\Omega$, a phenomenon related to the non-local character of the equation. Finally, we show that in a bounded domain, some regularization may occur, contrary to what happens in the whole space.

Article information

Source
Differential Integral Equations, Volume 20, Number 12 (2007), 1389-1404.

Dates
First available in Project Euclid: 20 December 2012

Permanent link to this document
https://projecteuclid.org/euclid.die/1356039071

Mathematical Reviews number (MathSciNet)
MR2377023

Zentralblatt MATH identifier
1211.47088

Subjects
Primary: 35K20: Initial-boundary value problems for second-order parabolic equations
Secondary: 35B05: Oscillation, zeros of solutions, mean value theorems, etc. 35D10

Citation

Chasseigne, Emmanuel. The Dirichlet problem for some nonlocal diffusion equations. Differential Integral Equations 20 (2007), no. 12, 1389--1404. https://projecteuclid.org/euclid.die/1356039071


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